Trapezoidal Rule of Integration

# Trapezoidal Rule of Integration - Chapter 07.02 Trapezoidal...

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Chapter 07.02 Trapezoidal Rule of Integration After reading this chapter, you should be able to: 1. derive the trapezoidal rule of integration, 2. use the trapezoidal rule of integration to solve problems, 3. derive the multiple-segment trapezoidal rule of integration, 4. use the multiple-segment trapezoidal rule of integration to solve problems, and 5. derive the formula for the true error in the multiple-segment trapezoidal rule of integration. What is integration? Integration is the process of measuring the area under a function plotted on a graph. Why would we want to integrate a function? Among the most common examples are finding the velocity of a body from an acceleration function, and displacement of a body from a velocity function. Throughout many engineering fields, there are (what sometimes seems like) countless applications for integral calculus. You can read about some of these applications in Chapters 07.00A-07.00G. Sometimes, the evaluation of expressions involving these integrals can become daunting, if not indeterminate. For this reason, a wide variety of numerical methods has been developed to simplify the integral. Here, we will discuss the trapezoidal rule of approximating integrals of the form ( 29 = b a dx x f I where ) ( x f is called the integrand, = a lower limit of integration = b upper limit of integration What is the trapezoidal rule? The trapezoidal rule is based on the Newton-Cotes formula that if one approximates the integrand by an th n order polynomial, then the integral of the function is approximated by 07.02.1

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07.02.2 Chapter 07.02 the integral of that th n order polynomial. Integrating polynomials is simple and is based on the calculus formula. Figure 1 Integration of a function 1 , 1 1 1 - + - = + + n n a b dx x n n b a n (1) So if we want to approximate the integral = b a dx x f I ) ( (2) to find the value of the above integral, one assumes ) ( ) ( x f x f n (3) where n n n n n x a x a x a a x f + + + + = - - 1 1 1 0 ... ) ( . (4) where ) ( x f n is a th n order polynomial. The trapezoidal rule assumes 1 = n , that is, approximating the integral by a linear polynomial (straight line), b a b a dx x f dx x f ) ( ) ( 1 Derivation of the Trapezoidal Rule Method 1: Derived from Calculus b a b a dx x f dx x f ) ( ) ( 1 + = b a dx x a a ) ( 1 0 - + - = 2 ) ( 2 2 1 0 a b a a b a (5)
Trapezoidal Rule 07.02.3 But what is 0 a and 1 a ? Now if one chooses, )) ( , ( a f a and )) ( , ( b f b as the two points to approximate ) ( x f by a straight line from a to b , a a a a f a f 1 0 1 ) ( ) ( + = = (6) b a a b f b f 1 0 1 ) ( ) ( + = = (7) Solving the above two equations for 1 a and 0 a , a b a f b f a - - = ) ( ) ( 1 a b a b f b a f a - - = ) ( ) ( 0 (8a) Hence from Equation (5), 2 ) ( ) ( ) ( ) ( ) ( ) ( 2 2 a b a b a f b f a b a b a b f b a f dx x f b a - - - + - - - (8b) + - = 2 ) ( ) ( ) ( b f a f a b (9) Method 2: Also Derived from Calculus ) ( 1 x f can also be approximated by using Newton’s divided difference polynomial as ) ( ) ( ) ( ) ( ) ( 1 a x a b a f b f a f x f - - - + = (10) Hence b a b a dx x f dx x f ) ( ) ( 1 - - - + = b a dx a x a b a f b f a f ) ( ) ( ) ( ) ( b a ax x a b a f b f x a f

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